{"schema_version":"claim-reader-v1","record":{"available_versions":[3,2,1],"current_version":3,"assessment_summary":"consistent","claim":{"statement":"If A is a complex abelian fourfold of Weil type with discriminant 1, the rational Hodge conjecture holds for A^m for every positive integer m.","id":"claim-hodge-weil-fourfold-powers","slug":"hodge-weil-fourfold-powers","version":1,"kind":"theorem","title":"All powers of Weil fourfolds with discriminant 1","assumptions":["A is a complex abelian fourfold of Weil type: an imaginary quadratic field acts with signature (2,2).","A has a compatible polarization of discriminant 1, in the paper’s Weil-type convention.","m is any positive integer; coefficients are rational."],"scope":"All codimensions on powers of this specified Weil-type family.","relation":"special_case","relationship_note":"For m=1 this overlaps Markman’s fourfold corollary; m>1 covers selected varieties of dimension 4m. 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Theorem 1.5.1","supersedes":null},{"id":"hodge-conjecture:hodge-weil-fourfold-powers:v1","source_version_id":"arxiv-2504.13607@arxiv-v2","item_id":"Theorem 1.2 and its proof, PDF pages 2 and 16","url":"https://arxiv.org/abs/2504.13607v2","wording":"If A is a complex abelian fourfold of Weil type with discriminant 1, the rational Hodge conjecture holds for A^m for every positive integer m.","wording_kind":"paraphrase","locator":"Theorem 1.2 and its proof, PDF pages 2 and 16","supersedes":null},{"id":"hodge-conjecture:hodge-withdrawal-2026:v1","source_version_id":"arxiv-2609.27916@arxiv-v2-withdrawn","item_id":"arXiv v2 comments and submission history","url":"https://arxiv.org/abs/2609.27916v2","wording":"A September 2026 manuscript claiming results on powers of CM fourfolds was withdrawn by its author on 8 October 2026 after reassessing its contribution.","wording_kind":"paraphrase","locator":"arXiv v2 comments and submission history","supersedes":null}],"sources":[{"work":{"authors":["Pierre Deligne"],"kind":"official_description","publication":"published","publication_note":"Official Clay problem description; 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For threefolds the curve-class case follows from hard Lefschetz and the divisor case; arbitrary fourfolds are excluded."},{"claim_id":"claim-hodge-abelian-fourfolds","aspect":"Algebraicity for families","summary":"Every abelian fourfold, including arbitrary Weil discriminants.","remaining":"Nonabelian fourfolds and arbitrary higher-dimensional abelian varieties remain outside this result.","overlaps_with":["claim-hodge-divisor-classes"],"overlap_note":"Corollary 1.6.1, using algebraicity of Weil classes and prior classification results. This does not prove the conjecture for arbitrary fourfolds or for all powers of every abelian fourfold."},{"claim_id":"claim-hodge-weil-fourfold-powers","aspect":"Algebraicity for families","summary":"Every power of a Weil-type fourfold with discriminant 1.","remaining":"Does not assert the all-powers statement for arbitrary discriminants or arbitrary abelian varieties.","overlaps_with":["claim-hodge-divisor-classes","claim-hodge-abelian-fourfolds"],"overlap_note":"For m=1 this overlaps Markman’s fourfold corollary; m>1 covers selected varieties of dimension 4m. The discriminant-1 hypothesis is retained for the all-powers assertion."},{"claim_id":"claim-hodge-motivated-transfer","aspect":"Conditional reductions","summary":"Hodge algebraicity transfers from all powers of a motivating variety.","remaining":"The motivating relation and the all-powers hypothesis are not automatically available for an arbitrary variety.","overlaps_with":["claim-hodge-weil-fourfold-powers"],"overlap_note":"Transfers established cases rather than resolving the unrestricted conjecture. Choosing a base with all-powers Hodge algebraicity turns the implication into an unconditional case."}],"activities":[{"id":"hodge-withdrawal-2026","title":"Recent source update · withdrawn preprint","kind":"source_correction","source_entry_ids":["hodge-conjecture:hodge-withdrawal-2026:v1"],"summary":"A September 2026 manuscript claiming results on powers of CM fourfolds was withdrawn by its author on 8 October 2026 after reassessing its contribution.","limitation":"The withdrawal is not a refutation of rational Hodge or of the other cited results. This manuscript contributes no additional proved progress in this record.","related_claim_ids":["claim-hodge-abelian-fourfolds","claim-hodge-weil-fourfold-powers"]}],"curation":{"checked_on":"2026-10-10","selection":"Selected classical cases, the 2025 abelian-fourfold result, a February 2026 revised proof for selected powers, one conditional transfer theorem and an integral-variant boundary.","search_scope":"Clay descriptions; author arXiv revisions and theorem statements; recent Hodge/abelian-fourfold literature; public Google DeepMind, Axiom, Harmonic and mathematical formalization search. Search cutoff: 10 October 2026. This selection is not exhaustive.","formal_status":"No complete Lean proof corresponding to these selected Hodge Claims has been established by this curation. 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It overlaps the low-dimensional theorem and every other indexed family in codimension one.","result":"proved","source_entry_ids":["hodge-conjecture:hodge-divisor-classes:v1"],"report_ids":["hodge-conjecture:hodge-divisor-classes:v1:report"],"formal_declarations":[],"independent_review":"pending"},{"statement":"The rational Hodge conjecture holds for every smooth projective complex variety of complex dimension at most three.","id":"claim-hodge-dimension-at-most-three","slug":"hodge-dimension-at-most-three","version":2,"kind":"theorem","title":"All codimensions in dimension ≤ 3","assumptions":["X is smooth and projective over C.","Complex dimension of X is at most 3.","Hodge classes and cycle combinations have rational coefficients."],"scope":"Every codimension, restricted to complex dimension ≤ 3.","relation":"special_case","relationship_note":"Includes the divisor case. For threefolds the curve-class case follows from hard Lefschetz and the divisor case; arbitrary fourfolds are excluded.","result":"proved","source_entry_ids":["hodge-conjecture:hodge-dimension-at-most-three:v2"],"report_ids":["hodge-conjecture:hodge-dimension-at-most-three:v2:report"],"formal_declarations":[],"independent_review":"pending"},{"statement":"The rational Hodge conjecture holds for every complex abelian variety of dimension four.","id":"claim-hodge-abelian-fourfolds","slug":"hodge-abelian-fourfolds","version":1,"kind":"theorem","title":"Abelian fourfolds · Markman (2025)","assumptions":["A is a complex abelian variety.","Complex dimension of A is 4.","Rational coefficients in cohomology and cycles."],"scope":"All codimensions on abelian fourfolds. No restriction to Weil discriminant 1 is needed for this corollary.","relation":"special_case","relationship_note":"Corollary 1.6.1, using algebraicity of Weil classes and prior classification results. 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The discriminant-1 hypothesis is retained for the all-powers assertion.","result":"proved","source_entry_ids":["hodge-conjecture:hodge-weil-fourfold-powers:v1"],"report_ids":["hodge-conjecture:hodge-weil-fourfold-powers:v1:report"],"formal_declarations":[],"independent_review":"pending"},{"statement":"If smooth projective complex varieties X and Y satisfy that Y is motivated by X in Arapura’s sense, and the rational Hodge conjecture holds on every power of X, it holds on every power of Y.","id":"claim-hodge-motivated-transfer","slug":"hodge-motivated-transfer","version":1,"kind":"theorem","title":"Conditional transfer through motivated varieties","assumptions":["X and Y are smooth projective over C.","Y is motivated by X in the category of André motives used in the paper, not merely weakly motivated.","Assume the rational Hodge conjecture for every positive power of X."],"scope":"A proved implication; its Hodge hypothesis must be supplied for the chosen X.","relation":"conditional_result","relationship_note":"Transfers established cases rather than resolving the unrestricted conjecture. Choosing a base with all-powers Hodge algebraicity turns the implication into an unconditional case.","result":"proved","source_entry_ids":["hodge-conjecture:hodge-motivated-transfer:v1"],"report_ids":["hodge-conjecture:hodge-motivated-transfer:v1:report"],"formal_declarations":[],"independent_review":"pending"},{"statement":"The integral analogue of the Hodge conjecture is false: there exist integral Hodge classes on smooth projective complex varieties not represented by integral algebraic cycles.","id":"claim-integral-hodge-counterexamples","slug":"integral-hodge-counterexamples","version":1,"kind":"conjecture","title":"Integral Hodge variant · counterexamples","assumptions":["Integral, rather than rational, coefficients.","Smooth projective complex varieties; the assertion is existential."],"scope":"Atiyah–Hirzebruch counterexamples to the integral variant; not a counterexample to rational Hodge.","relation":"related_variant","relationship_note":"Changing the coefficient ring changes the statement. Rational Hodge stays open; this refutation must never label the canonical Problem refuted.","result":"refuted","source_entry_ids":["hodge-conjecture:integral-hodge-counterexamples:v1"],"report_ids":["hodge-conjecture:integral-hodge-counterexamples:v1:report"],"formal_declarations":[],"independent_review":"pending"}],"problem":{"resolution_scope":{"resolution":"unresolved","extent":"general_case","summary":"Divisors, varieties of dimension at most three, abelian fourfolds and selected powers are covered by cited theorems; conditional transfer results are also indexed.","report_ids":["hodge-conjecture:report:0:v2"],"progress_claim_ids":["claim-hodge-divisor-classes","claim-hodge-dimension-at-most-three","claim-hodge-abelian-fourfolds","claim-hodge-weil-fourfold-powers","claim-hodge-motivated-transfer"],"remaining":"Arbitrary smooth projective varieties remain open, including general fourfolds in codimension two. Abelian fourfolds are a solved family, not all fourfolds."},"statement_kind":"conjecture","problem_id":"problem-hodge-conjecture","slug":"hodge-conjecture","title":"Hodge conjecture","question":"On a smooth complex projective variety, is every rational Hodge class a rational linear combination of classes of algebraic cycles?","scope":"Rational Hodge classes on smooth projective varieties over C, in every codimension. Integral coefficients, nonprojective spaces, the generalized Hodge conjecture and the Tate conjecture are distinct variants.","aliases":["Hodge conjecture"],"fields":["Algebraic geometry"],"msc":["14C30"],"topics":["Scoped progress","Conditional results"],"background":"specialist","scale":"major conjecture","depth":"source-linked","source_entry_ids":["hodge-conjecture:entry:0:v1","hodge-conjecture:entry:0:v2","hodge-conjecture:hodge-divisor-classes:v1","hodge-conjecture:hodge-dimension-at-most-three:v2","hodge-conjecture:hodge-abelian-fourfolds:v1","hodge-conjecture:hodge-weil-fourfold-powers:v1","hodge-conjecture:hodge-motivated-transfer:v1","hodge-conjecture:integral-hodge-counterexamples:v1","hodge-conjecture:hodge-withdrawal-2026:v1"],"status_report_ids":["hodge-conjecture:report:0:v1","hodge-conjecture:report:0:v2"],"editorial":null,"literature":[{"title":"Clay problem page and official description","url":"https://www.claymath.org/millennium/hodge-conjecture/","note":"Clay reports the general question unsolved, observed 10 October 2026. This current-state observation is separate from the dated theorem and artifact citations below."}],"formalizations":[],"related":[],"tasks":[],"insights":[],"coverage_gaps":["Selected results, not an exhaustive literature survey or a percentage of the conjecture solved.","Source-attributed theorem statements; independent mathematical and relationship review is pending.","External papers, programs and Lean placeholders have no OpenMath execution or independent reproduction attached.","AI involvement is recorded only where the cited source reports it; absence from this selection does not establish absence of work."],"checked_on":"2026-10-10"},"sources":[{"id":"clay-millennium@observed-2026-10-01","source_id":"clay-millennium","title":"Clay · Millennium Problems","owner":"Clay Mathematics Institute","url":"https://www.claymath.org/millennium-problems/","subject_scope":"Seven Millennium Prize Problems; five selected here","revision":"observed-2026-10-01","content_sha256":null,"update_method":"Manual checked import; new observations require staff preview and publication","identifier_policy":"Canonical source URL plus explicit item/declaration locator; no fuzzy identity merge","observed_on":"2026-10-01","source_updated_on":null,"rights":"metadata-link","license":"Not established; links and OpenMath paraphrases only","terms_url":"https://www.claymath.org/millennium-problems/","attribution":"Clay Mathematics Institute","rights_note":"Store metadata and original OpenMath paraphrases; follow the original source for authoritative wording.","check_interval_days":30},{"id":"clay-millennium@observed-2026-10-10","source_id":"clay-millennium","title":"Clay · Millennium Problems","owner":"Clay Mathematics Institute","url":"https://www.claymath.org/millennium-problems/","subject_scope":"Seven Millennium Prize Problems; five selected here","revision":"observed-2026-10-10","content_sha256":null,"update_method":"Manual checked import; new observations require staff preview and publication","identifier_policy":"Canonical source URL plus explicit item/declaration locator; 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recheck upstream before a new publication.","identifier_policy":"Stable bibliographic identifier and explicit revision; theorem locators belong to source entries.","observed_on":"2026-10-10","source_updated_on":null,"rights":"metadata-link","license":"Links and original OpenMath paraphrases only; upstream license governs external files.","terms_url":"https://www.claymath.org/wp-content/uploads/2022/06/hodge.pdf","attribution":"Pierre Deligne","rights_note":"No upstream full text or code is republished by this record.","check_interval_days":90},{"work":{"authors":["Eyal Markman"],"kind":"research_paper","publication":"preprint","publication_note":"Author preprint at the cited revision; journal acceptance has not been established by this curation.","ai_role":"not_reported","ai_note":"The cited work does not report AI involvement; this is not an assertion that no AI was used.","artifacts":[{"kind":"pdf","title":"Proof PDF · v2","url":"https://arxiv.org/pdf/2502.03415v2","version":"arXiv:2502.03415v2","assessment":"paper_proof","note":"External evidence linked; no OpenMath execution or independent reproduction is asserted."}]},"id":"arxiv-2502.03415@arxiv-v2","source_id":"arxiv-2502.03415","title":"Markman · cycles from secant sheaves","owner":"Eyal Markman","url":"https://arxiv.org/abs/2502.03415v2","subject_scope":"Markman · cycles from secant sheaves","revision":"arxiv-v2","content_sha256":null,"update_method":"Manual source and revision check; recheck upstream before a new publication.","identifier_policy":"Stable bibliographic identifier and explicit revision; theorem locators belong to source entries.","observed_on":"2026-10-10","source_updated_on":"2025-06-08","rights":"metadata-link","license":"Links and original OpenMath paraphrases only; upstream license governs external files.","terms_url":"https://arxiv.org/abs/2502.03415v2","attribution":"Eyal Markman","rights_note":"No upstream full text or code is republished by this record.","check_interval_days":90},{"work":{"authors":["Salvatore Floccari","Lie Fu"],"kind":"research_paper","publication":"accepted","publication_note":"arXiv v2 states final version, to appear in Journal de Mathématiques Pures et Appliquées; 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upstream license governs external files.","terms_url":"https://arxiv.org/abs/2504.13607v2","attribution":"Salvatore Floccari; Lie Fu","rights_note":"No upstream full text or code is republished by this record.","check_interval_days":90},{"work":{"authors":["Donu Arapura"],"kind":"research_paper","publication":"published","publication_note":"Published in Advances in Mathematics 207 (2006), 762–781; author version cited.","ai_role":"not_reported","ai_note":"The cited work does not report AI involvement; this is not an assertion that no AI was used.","artifacts":[{"kind":"pdf","title":"Author proof PDF · v3","url":"https://arxiv.org/pdf/math/0501348v3","version":"arXiv:math/0501348v3","assessment":"paper_proof","note":"External evidence linked; no OpenMath execution or independent reproduction is asserted."}]},"id":"arxiv-math-0501348@arxiv-v3","source_id":"arxiv-math-0501348","title":"Arapura · motivation for Hodge cycles","owner":"Donu Arapura","url":"https://arxiv.org/abs/math/0501348v3","subject_scope":"Arapura · motivation for Hodge cycles","revision":"arxiv-v3","content_sha256":null,"update_method":"Manual source and revision check; recheck upstream before a new publication.","identifier_policy":"Stable bibliographic identifier and explicit revision; theorem locators belong to source entries.","observed_on":"2026-10-10","source_updated_on":"2005-12-16","rights":"metadata-link","license":"Links and original OpenMath paraphrases only; upstream license governs external files.","terms_url":"https://arxiv.org/abs/math/0501348v3","attribution":"Donu Arapura","rights_note":"No upstream full text or code is republished by this record.","check_interval_days":90},{"work":{"authors":["Ningyi Li"],"kind":"research_paper","publication":"withdrawn","publication_note":"Author withdrew v2 on 8 October 2026 after reassessing its contribution; no standalone publication planned.","ai_role":"not_reported","ai_note":"The cited work does not report AI involvement; this is not an assertion that no AI was used.","artifacts":[{"kind":"paper","title":"Withdrawal notice · v2","url":"https://arxiv.org/abs/2609.27916v2","version":"arXiv:2609.27916v2","assessment":"external_report","note":"External evidence linked; no OpenMath execution or independent reproduction is asserted."}]},"id":"arxiv-2609.27916@arxiv-v2-withdrawn","source_id":"arxiv-2609.27916","title":"Li · withdrawn CM-fourfold powers preprint","owner":"Ningyi Li","url":"https://arxiv.org/abs/2609.27916v2","subject_scope":"Li · withdrawn CM-fourfold powers preprint","revision":"arxiv-v2-withdrawn","content_sha256":null,"update_method":"Manual source and revision check; recheck upstream before a new publication.","identifier_policy":"Stable bibliographic identifier and explicit revision; theorem locators belong to source entries.","observed_on":"2026-10-10","source_updated_on":"2026-10-08","rights":"metadata-link","license":"Links and original OpenMath paraphrases only; upstream license governs external files.","terms_url":"https://arxiv.org/abs/2609.27916v2","attribution":"Ningyi Li","rights_note":"No upstream full text or code is republished by this record.","check_interval_days":90}],"entries":[{"id":"hodge-conjecture:entry:0:v1","source_version_id":"clay-millennium@observed-2026-10-01","item_id":"https://www.claymath.org/millennium/hodge-conjecture/","url":"https://www.claymath.org/millennium/hodge-conjecture/","wording":"On a smooth complex projective variety, is every rational Hodge class a rational linear combination of classes of algebraic cycles?","wording_kind":"paraphrase","locator":"https://www.claymath.org/millennium/hodge-conjecture/","supersedes":null},{"id":"hodge-conjecture:entry:0:v2","source_version_id":"clay-millennium@observed-2026-10-10","item_id":"https://www.claymath.org/millennium/hodge-conjecture/","url":"https://www.claymath.org/millennium/hodge-conjecture/","wording":"On a smooth complex projective variety, is every rational Hodge class a rational linear combination of classes of algebraic cycles?","wording_kind":"paraphrase","locator":"https://www.claymath.org/millennium/hodge-conjecture/","supersedes":"hodge-conjecture:entry:0:v1"},{"id":"hodge-conjecture:hodge-divisor-classes:v1","source_version_id":"clay-deligne-hodge@official-description","item_id":"Section 2(iii), PDF page 2","url":"https://www.claymath.org/wp-content/uploads/2022/06/hodge.pdf","wording":"For a smooth projective complex variety X, every rational class in H²(X,Q) of type (1,1) is a rational combination of divisor classes.","wording_kind":"paraphrase","locator":"Section 2(iii), PDF page 2","supersedes":null},{"id":"hodge-conjecture:hodge-dimension-at-most-three:v2","source_version_id":"clay-millennium@observed-2026-10-10","item_id":"Clay Hodge page, overview: dimension less than four","url":"https://www.claymath.org/millennium/hodge-conjecture/","wording":"The rational Hodge conjecture holds for every smooth projective complex variety of complex dimension at most three.","wording_kind":"paraphrase","locator":"Clay Hodge page, overview: dimension less than four","supersedes":null},{"id":"hodge-conjecture:hodge-abelian-fourfolds:v1","source_version_id":"arxiv-2502.03415@arxiv-v2","item_id":"Corollary 1.6.1, PDF page 9; Theorem 1.5.1","url":"https://arxiv.org/abs/2502.03415v2","wording":"The rational Hodge conjecture holds for every complex abelian variety of dimension four.","wording_kind":"paraphrase","locator":"Corollary 1.6.1, PDF page 9; Theorem 1.5.1","supersedes":null},{"id":"hodge-conjecture:hodge-weil-fourfold-powers:v1","source_version_id":"arxiv-2504.13607@arxiv-v2","item_id":"Theorem 1.2 and its proof, PDF pages 2 and 16","url":"https://arxiv.org/abs/2504.13607v2","wording":"If A is a complex abelian fourfold of Weil type with discriminant 1, the rational Hodge conjecture holds for A^m for every positive integer m.","wording_kind":"paraphrase","locator":"Theorem 1.2 and its proof, PDF pages 2 and 16","supersedes":null},{"id":"hodge-conjecture:hodge-motivated-transfer:v1","source_version_id":"arxiv-math-0501348@arxiv-v3","item_id":"Lemma 4.2, PDF pages 9–10","url":"https://arxiv.org/abs/math/0501348v3","wording":"If smooth projective complex varieties X and Y satisfy that Y is motivated by X in Arapura’s sense, and the rational Hodge conjecture holds on every power of X, it holds on every power of Y.","wording_kind":"paraphrase","locator":"Lemma 4.2, PDF pages 9–10","supersedes":null},{"id":"hodge-conjecture:integral-hodge-counterexamples:v1","source_version_id":"clay-deligne-hodge@official-description","item_id":"Section 2(iv), PDF page 2","url":"https://www.claymath.org/wp-content/uploads/2022/06/hodge.pdf","wording":"The integral analogue of the Hodge conjecture is false: there exist integral Hodge classes on smooth projective complex varieties not represented by integral algebraic cycles.","wording_kind":"paraphrase","locator":"Section 2(iv), PDF page 2","supersedes":null},{"id":"hodge-conjecture:hodge-withdrawal-2026:v1","source_version_id":"arxiv-2609.27916@arxiv-v2-withdrawn","item_id":"arXiv v2 comments and submission history","url":"https://arxiv.org/abs/2609.27916v2","wording":"A September 2026 manuscript claiming results on powers of CM fourfolds was withdrawn by its author on 8 October 2026 after reassessing its contribution.","wording_kind":"paraphrase","locator":"arXiv v2 comments and submission history","supersedes":null}],"reports":[{"id":"hodge-conjecture:report:0:v1","entry_id":"hodge-conjecture:entry:0:v1","state":"reported_open","source_label":"Unsolved","scope":"The cited source item and version only; this is not an independent OpenMath assessment.","source_date":null,"observed_on":"2026-10-01","evidence_url":"https://www.claymath.org/millennium/hodge-conjecture/","supersedes":null},{"id":"hodge-conjecture:report:0:v2","entry_id":"hodge-conjecture:entry:0:v2","state":"reported_open","source_label":"Unsolved","scope":"The cited source item and version only; this is not an independent OpenMath assessment.","source_date":null,"observed_on":"2026-10-10","evidence_url":"https://www.claymath.org/millennium/hodge-conjecture/","supersedes":"hodge-conjecture:report:0:v1"},{"id":"hodge-conjecture:hodge-divisor-classes:v1:report","entry_id":"hodge-conjecture:hodge-divisor-classes:v1","state":"resolved","source_label":"Divisor classes · Lefschetz (1,1)","scope":"Codimension one in any dimension; rational coefficients.","source_date":null,"observed_on":"2026-10-10","evidence_url":"https://www.claymath.org/wp-content/uploads/2022/06/hodge.pdf","supersedes":null},{"id":"hodge-conjecture:hodge-dimension-at-most-three:v2:report","entry_id":"hodge-conjecture:hodge-dimension-at-most-three:v2","state":"resolved","source_label":"All codimensions in dimension ≤ 3","scope":"Every codimension, restricted to complex dimension ≤ 3.","source_date":null,"observed_on":"2026-10-10","evidence_url":"https://www.claymath.org/millennium/hodge-conjecture/","supersedes":null},{"id":"hodge-conjecture:hodge-abelian-fourfolds:v1:report","entry_id":"hodge-conjecture:hodge-abelian-fourfolds:v1","state":"resolved","source_label":"Abelian fourfolds · Markman (2025)","scope":"All codimensions on abelian fourfolds. No restriction to Weil discriminant 1 is needed for this corollary.","source_date":"2025-06-08","observed_on":"2026-10-10","evidence_url":"https://arxiv.org/abs/2502.03415v2","supersedes":null},{"id":"hodge-conjecture:hodge-weil-fourfold-powers:v1:report","entry_id":"hodge-conjecture:hodge-weil-fourfold-powers:v1","state":"resolved","source_label":"All powers of Weil fourfolds with discriminant 1","scope":"All codimensions on powers of this specified Weil-type family.","source_date":"2026-02-10","observed_on":"2026-10-10","evidence_url":"https://arxiv.org/abs/2504.13607v2","supersedes":null},{"id":"hodge-conjecture:hodge-motivated-transfer:v1:report","entry_id":"hodge-conjecture:hodge-motivated-transfer:v1","state":"resolved","source_label":"Conditional transfer through motivated varieties","scope":"A proved implication; its Hodge hypothesis must be supplied for the chosen X.","source_date":"2005-12-16","observed_on":"2026-10-10","evidence_url":"https://arxiv.org/abs/math/0501348v3","supersedes":null},{"id":"hodge-conjecture:integral-hodge-counterexamples:v1:report","entry_id":"hodge-conjecture:integral-hodge-counterexamples:v1","state":"refuted","source_label":"Integral Hodge variant · counterexamples","scope":"Atiyah–Hirzebruch counterexamples to the integral variant; not a counterexample to rational Hodge.","source_date":null,"observed_on":"2026-10-10","evidence_url":"https://www.claymath.org/wp-content/uploads/2022/06/hodge.pdf","supersedes":null}]},"status":{"state":"open","label":"Open","extent":"general_case","summary":"Divisors, varieties of dimension at most three, abelian fourfolds and selected powers are covered by cited theorems; conditional transfer results are also indexed.","basis":"source","attributed_to":"Clay · Millennium Problems","as_of":"2026-10-10","progress_label":"Special cases & conditional results","progress_claim_ids":["claim-hodge-divisor-classes","claim-hodge-dimension-at-most-three","claim-hodge-abelian-fourfolds","claim-hodge-weil-fourfold-powers","claim-hodge-motivated-transfer"],"remaining":"Arbitrary smooth projective varieties remain open, including general fourfolds in codimension two. Abelian fourfolds are a solved family, not all fourfolds.","views":["open","partial"]}},"selected_context":null}}